On Big Ramsey degrees of universal $\omega$-edge-labeled hypergraphs

We show that the big Ramsey degrees of every countable universal $u$-uniform $\omega$-edge-labeled hypergraph are infinite for every $u\ge 2$. Together with a recent result of Braunfeld, Chodounský, de Rancourt, Hubička, Kawach, and Konečný this finishes full characterisation of unrestricted relational structures with finite big Ramsey degrees.